| L(s) = 1 | + 2.73·2-s + 5.46·4-s − 5-s − 1.26·7-s + 9.46·8-s − 2.73·10-s + 2.26·11-s − 5.46·13-s − 3.46·14-s + 14.9·16-s − 0.732·17-s − 2.46·19-s − 5.46·20-s + 6.19·22-s − 3.46·23-s + 25-s − 14.9·26-s − 6.92·28-s + 7.19·29-s − 3·31-s + 21.8·32-s − 2·34-s + 1.26·35-s + 0.732·37-s − 6.73·38-s − 9.46·40-s − 3.19·41-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.73·4-s − 0.447·5-s − 0.479·7-s + 3.34·8-s − 0.863·10-s + 0.683·11-s − 1.51·13-s − 0.925·14-s + 3.73·16-s − 0.177·17-s − 0.565·19-s − 1.22·20-s + 1.32·22-s − 0.722·23-s + 0.200·25-s − 2.92·26-s − 1.30·28-s + 1.33·29-s − 0.538·31-s + 3.86·32-s − 0.342·34-s + 0.214·35-s + 0.120·37-s − 1.09·38-s − 1.49·40-s − 0.499·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.759431297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.759431297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 0.732T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 + 3.26T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 0.267T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 7.66T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.81280704555777175007360061082, −10.70111419310084895450262454155, −9.764901978379504904420787430943, −8.118839625677205964674179257176, −6.98720415408259830572734549482, −6.44725846206581020854653092934, −5.20184647625128074553158387134, −4.36172788927928137817911118524, −3.39185724954111398066867151906, −2.22456788844258673958989510895,
2.22456788844258673958989510895, 3.39185724954111398066867151906, 4.36172788927928137817911118524, 5.20184647625128074553158387134, 6.44725846206581020854653092934, 6.98720415408259830572734549482, 8.118839625677205964674179257176, 9.764901978379504904420787430943, 10.70111419310084895450262454155, 11.81280704555777175007360061082
